Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

272system of seven equations:(8.27)0 = −Yx 2 + (2x + g1 )Yx 1 ,0 = −Y 2y − (2x + 1 g1 )Y 2y + (2 + 2 g1 x )X + (2 + 2g2 )Yx 1 + (2x + g1 )Y 1y 1++ s X + r Y 1 + r Y 2 + s Yx 1 + k ∗ [2x + g 1 ] 2 X x ,0 = −Y 2y − X 2 x + 2 Y 1y + s X + r Y 1 + r Y 2 + s Y 11 x + k∗ (6x + 3g 1 )X x ,0 = −X y 1 + r X + r Y 1 + r Y 2 + s Yx 1 + s X x + s Y 1y + s Y 2 1 y 2,0 = −X y 2 + r X + r Y 1 + r Y 2 + s Yx 1 + s X x + s Y 1y + s Y 2 1 y 2,0 = −Yxx 1 ,0 = −2 Y 1xy + X ( 1 xx 1 − k ∗ (4x + 2g 1 ) ) + r X + r Y 1 + r Y 2 + r Yx 1 . + s X x.Restarting from this system, we differentiate (8.27) 3 with respect to x:(8.28)0 = −Y 2xy − X 2 xx + 2 Y 1xy + r X + r Y 1 + r Y 2 +1+ r X x + r Yx 1 + r Yx 2 + s Yxx 1 + k ∗ (6x + 3g 1 )X xx .We replace Yx 21, we erase Yxx and we add (8.27) 7:(8.29) 0 = −Y 2xy 2 +k∗ (2x+g 1 )X xx +r X +r Y 1 +r Y 2 +r Yx 1 +r X x.We differentiate (8.27) 2 with respect to x:(8.30)0 = −Y 2xy − (2 + 1 g1 x )Y 2y − (2x + 2 g1 )Y 2xy + r X + (2 + 2 g1 x )X x++ s Yx 1 + (2 + 2g2 )Yxx 1 + (2 + g1 x )Y 1y + (2x + 1 g1 )Y 1xy + r X x+1+ s X x + r Y 1 + r Yx 1 + r Y 2 + r Yx 2 + r Yx 1 + s Yxx 1 + k ∗ [2x + g 1 ] 2 X xx .Differentiating (8.27) 1 with respect to y 1 , we may substract 0 = −Y 2xy+ 1(2x + g 1 )Y 1xy+ r Y 1 1 x ; we replace Yx 2 and erase Yxx; 1 we substract (8.29)multiplied by (2x + g 1 ); we get:(8.31) 0 = −Y 2y 2 + (1 + s)X x + Y 1y 1 + r X + r Y 1 + r Y 2 + r Y 1x .Comparing with (8.27) 3 yields:Y 1y(8.32)= (2 + s)X 1 x + r X + r Y 1 + r Y 2 + r Yx 1 ,Y 2y = (3 + s)X 2 x + r X + r Y 1 + r Y 2 + r Yx 1 .These are (8.18) 5 and (8.18) 11 . Differentiating these two equations withrespect to x, replacing Yx 21and erasing Yxx , we get:Y 1xy(8.33)= (2 + s)X 1 xx + r X + r Y 1 + r Y 2 + r Yx 1 + r X x,Y 2xy = (3 + s)X 2 xx + r X + r Y 1 + r Y 2 + r Yx 1 + r X x .We then replace this value of Y 2xy 2 in (8.29) and solve X xx : this yields(8.18) 2 .